Financial Mathematics for Actuaries (M339W=M389W)

    Important external links
  • Link to the Sample SoA problems and solutions - Part I
  • MFE Exam, Spring 2007: Problems
  • MFE Exam, Spring 2007: Solutions
  • MFE Exam, Spring 2009: Problems
  • MFE Exam, Spring 2009: Solutions
  • Link to the Measures of Investment Risk, Monte Carlo Simulation, and Empirical Evidence on the Efficient Markets Hypothesis.
  • Link to the Actuarial Applications of Options and Other Financial Derivatives.
  • Link to the Sample SoA problems and solutions - Part II
  • Link to the IFM exam tables




  • Class 1: January 22nd
    The binomial asset pricing model [review].
  • First-Day Handout
  • Lecture Sixteen: The binomial asset-pricing model.
  • Class notes: Review of the binomial asset pricing model.



  • Class 2: January 24th
    Binomial option pricing [review].
  • Lecture Seventeen: Binomial option pricing.
  • Lecture Twelve: Gap options
  • Lecture Thirteen: Exchange options
  • Problem set #1.
  • Class notes: Review of the binomial option pricing.
  • Last term's prerequisite in-term -- Problems
  • Last term's prerequisite in-term -- Solutions
  • Homework #1 -- due on Friday, January 31st
  • Quiz #1 -- due on Monday, January 27th



  • Class 3: January 27th
    Binomial pricing of currency options. Binomial pricing of futures options.
  • Class notes: Currency and futures options (the binomial model).
  • Quiz #1 -- Solutions
  • Quiz #2 -- due on Wednesday, January 29th



  • Class 4: January 29th
    More on futures options. Subjective expectations and forward contracts.
  • Class notes: Futures options (the binomial model). Subjective expectations.
  • Class notes: Subjective expectations. Realized returns.
  • Quiz #2 -- Solutions



  • Class 5: January 31st
    Subjective expectations and forward contracts[cont'd].
  • Problem set #2
  • Scaled random walk. Central limit theorem.
  • Mathematica Code: One Random Walk
  • YouTube: One Random Walk
  • Mathematica Code: Binomial model limit
  • (In-class) Warm-up #1 -- Solutions
  • Homework #1 -- Solutions



  • Class 6: February 3rd
  • The prerequisite in-term exam
  • The prerequisite in-term exam -- Solutions



  • Class 7: February 5th
    Realized returns.
  • Class notes: Realized returns.



  • Class 8: February 7th
    The log-normal distribution. Jensen's inequality.
  • Class notes: The log-normal distribution. Jensen's inequality.
  • The Log Normal Distribution from the Wolfram Demonstrations Project by Chris Boucher
  • Mathematica demonstration: Modeling return distributions



  • Class 9: February 10th
    Log-normal stock prices. Parameter interpretation.
  • Class notes: Lognormal stock prices: Mean. Median.
  • Problem set #3
  • Quiz #3 -- due on Friday, February 14th



  • Class 10: February 12th
    Measures of risk: variance, semi-variance, VaR.
  • Wikipedia: Variance. Semi-variance.
  • Wikipedia: Value at Risk (VaR).
  • Class notes: Risk measures: Variance. Semi-variance. Value-at-Risk.



  • Class 11: February 14th
    Log-normal stock prices: Tail probabilities.
  • Class notes: Tail probabilities.
  • Problem set #4 -- Solutions
  • Quiz #3 -- Solutions
  • Quiz #4 -- due on Wednesday, February 19th
  • Homework #2 -- due on Monday, February 24th



  • Class 12: February 17th
    Lognormal "confidence" intervals. VaR (revisited).
  • Class notes: Lognormal "confidence" intervals. VaR (revisited).



  • Class 13: February 19th
    TVaR. Coherent risk measures.
  • Class notes: Tail Value-at-Risk (TVaR). Coherent risk measures.
  • Wikipedia: Tail Value at Risk (TVaR).
  • Wikipedia: Coherent risk measures.
  • Quiz #4 -- Solutions



  • Class 14: February 21st
    Log-normal partial and conditional expectations. Black-Scholes prices.
  • Class notes: Log-normal partial and conditional expectations. Black-Scholes prices.
  • Wikipedia: Black-Scholes
  • The original article by Black and Scholes
  • Mathematica Code: Black-Scholes price
  • Quiz #5 -- due on Wednesday, February 26th



  • Class 15: February 24th
    Black-Scholes prices [cont'd].
  • Class notes: Black-Scholes pricing: Calls. Puts. Forward start options.
  • Black-Scholes prices [cont'd]. The "master" Black-Scholes pricing formula.
  • Class notes: Black-Scholes pricing: Rolling insurance strategy. Gap options. The "master" Black-Scholes pricing formula.
  • Problem set #5 -- Solutions
  • Homework #2 -- Solutions



  • Class 16: February 26th
    Black-Scholes prices: Currency options. Futures options.
  • Class notes: Black-Scholes pricing: Currency options. Futures options.
  • Quiz #5 -- Solutions



  • Class 17: February 28th
    Black-Scholes prices: Discrete-dividend-paying stocks.
  • Class notes: Black-Scholes pricing: Discrete-dividend-paying stocks.
  • Quiz #6 -- due on Monday, March 2nd



  • Class 18: March 2nd
    Black-Scholes prices: Discrete-dividend-paying stocks [cont'd].
    Option Greeks in the Black-Scholes model.
  • Wikipedia: Greeks
  • Class notes: Discrete-dividend stocks. Black-Scholes greeks.
  • Slides: Option Greeks.
  • In-Term One -- practice problems
  • Sample In-Term One -- Solutions
  • Quiz #6 -- Solutions



  • Class 19: March 4th
    Focus on the delta.
  • Class notes: Focus on the delta



  • Class 20: March 6th
    Option elasticity.
  • Class notes: Option elasticity.
  • Suggested Problem: MFE Exam Spring 2007: Problem #5 (option volatility)
  • Homework #3 -- due on Monday, March 30th



  • Class 21: March 9th
  • In-term exam I
  • In-Term One -- Solutions



  • Class 22: March 11th
    Focus on gamma. Other greeks.
  • Class notes: Gamma.
  • Mathematica Code: Black-Scholes Greeks
  • YouTube: Black-Scholes Greeks



  • Class 23: March 13th
    UT closed. No classes due to COVID-19.
  • The updated First-Day Handout



  • Class 24: March 23rd
    No classes due to COVID-19.



    Class 25: March 25th
    No classes due to COVID-19.



    Class 26: March 27th
    No classes due to COVID-19.



    Class 27: March 30th
    Getting used to Zoom.
  • Homework #3 -- Solutions



  • Class 28: April 1st
    Implied volatility.
  • Wikipedia: Implied volatility.
  • Class notes: Implied volatility. Delta-gamma-theta approximation.
  • YouTube: Implied volatility.
  • Homework #4 -- due on Wednesday, April 8th



  • Class 29: April 3rd
    Delta-gamma-theta approximation.
  • Class notes: Implied volatility. Delta-gamma-theta approximation.
  • Market-making and delta-hedging.
  • Wikipedia: Market maker.
  • Class notes: Market making. Delta hedging.
  • Slides: Market making. Delta hedging.



  • Class 30: April 6th
    Delta-hedger's profit.
  • Class notes: Delta-hedger's profit.
  • Quiz #10 -- due on Friday, April 10th



  • Class 31: April 8th
    Delta-gamma hedging.
  • Class notes: Delta-gamma hedging.
  • Homework #4 -- Solutions
  • Quiz #11 -- due on Monday, April 13th
  • In-Term Two -- Topics
  • Sample In-Term Two -- Problems
  • Sample In-Term Two -- Solutions



  • Class 32: April 10th
    Exchange options.
  • Class notes: Exchange options.
  • Quiz #10 -- Solutions
  • Quiz #12 -- due on Friday, April 17th
  • Homework #5 -- due on Monday, April 20th



  • Class 33: April 13th
    Analyzing the project. Break-even analysis.
  • Wikipedia: Cost of capital.
  • Wikipedia: Equity cost of capital.
  • Wikipedia: Capital budgeting.
  • Planet money: Spreadsheets!
  • Class notes: Break-even analysis.
  • Quiz #13 -- due on Wednesday, April 22nd
  • Information, competition, and stock prices. The efficient market hypothesis. Historical returns of stocks and bonds. Confidence intervals for the mean return.
  • Wikipedia: Efficient-market hypothesis.
  • Class notes: Information, competition, and stock prices.
  • Quiz #11 -- Solutions
  • Mock In-Term Two -- Problems
  • Mock In-Term Two -- Solutions



  • Class 34: April 15th
  • In-term exam II
  • In-Term Two -- Solutions



  • Class 35: April 17th
    The expected return of a portfolio. The volatility of a two-stock portfolio.
    The volatility of a large portfolio.
  • Class notes: The expected return of a portfolio. The volatility of a two-stock portfolio. The volatility of a large portfolio.
  • Quiz #12 -- Solutions



  • Class 36: April 20th
    Diversification. Choosing an efficient portfolio.
  • Class notes: Diversification.
  • Homework #5 -- Solutions
  • Quiz #14 -- due on Friday, April 24th



  • Class 37: April 22nd
    Feasible sets. Effect of correlation.
  • Mathematica demonstration: A Two-Asset Markowitz Feasible Set
  • Mathematica demonstration: A Three-Asset Feasible Set and Efficient Frontier
  • Class notes: Feasible sets. Effect of correlation. Sharpe ratio.
  • Quiz #13 -- Solutions
  • Homework #6 -- due on Wednesday, April 29th



  • Class 38: April 24th
    Sharpe ratio.
  • Class notes: Required returns.
  • Quiz #14 -- Solutions
  • Quiz #15 -- due on Monday, April 27th



  • Class 39: April 27th
    Required returns. Betas of stocks. The Capital Asset Pricing Model (CAPM): Assumptions.
  • Class notes: Required returns.
  • Wikipedia: CAPM
  • Quiz #15 -- Solutions
  • Quiz #16 -- due on Friday, May 1st



  • Class 40: April 29th
    CAPM: Consequences.
  • Class notes: CAPM: Consequences.
  • Quiz #17 -- due on Monday, May 4th
  • Homework #6 -- Solutions
  • Homework #7 -- due on Wednesday, May 6th



  • Class 41: May 1st
    Alphas of stocks.
  • Class notes: Alphas of stocks.
  • Wikipedia: Linear regression.
  • Quiz #16 -- Solutions



  • Class 42: May 4th
    Behavior of individual investors. Systematic trading biases.
  • Class notes: Behavior of individual investors.
  • Quiz #17 -- Solutions
  • Quiz #18 -- due on Friday, May 8th
  • An article on familiarity bias.
  • Familiarity breeds investment.
  • Wikipedia: Overconfidence effect.
  • Wikipedia: Disposition effect.
  • Investopedia: Herd behavior.



  • Class 43: May 6th
    The efficient market hypothesis.
  • Wikipedia: Efficient-market hypothesis.
  • Multifactor models.
  • Class notes: Multifactor models.
  • Class notes: Multifactor models[cont'd].
  • Wikipedia: Multifactor models.
  • Wikipedia: The Fama-French model.
  • Wikipedia: The Carhart four-factor model.
  • Homework #7 -- Solutions
  • Quiz #19 -- with solutions, not to be handed in
  • Real options.
  • Investopedia: Real options.



  • Class 44: May 8th
    SLLN. Monte Carlo simulation.
  • Wikipedia: Random number generation
  • Wikipedia: Mersenne twister
  • Mathematica Code: Unit Uniform
  • YouTube: Unit uniform
  • Slides: Probability on Cointoss Space
  • Class notes: Binomial stock-price trees with many periods. SLLN. CLT
  • Law of large numbers. Monte carlo pricing (binomial).
  • Class notes: SLLN. Monte Carlo.
  • Mathematica Code: Binomial Monte Carlo
  • Quiz #18 -- Solutions